Configurations in Fractals

On polynomial configurations in fractal sets

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.

分散型Furstenberg集II：一般构型与分形限制投影

Spread-out Furstenberg sets II: general configurations and fractal restricted projections

该论文针对阴影未局部集中于 pencil 的特定构型证明最优对偶 Furstenberg 估计，推导平面及高维限制投影定理，并将其应用于 Grassmann 流形曲面投影、固定距离等多个分形相关问题。

Patterns in Random Fractals

We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemerédi theorems for random discrete sets, we also consider the corresponding problem for sets of positive $ν$-measure, where $ν$ is the natural measure on $A$. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating t…

Finite Point Configurations and the Regular Value Theorem in a Fractal setting

In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $E\subset \mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the finite point configuration set depends on that of $E$. In particular, we show that if a planar set has dimension exceeding $\frac{5}{4}$, then there exists a point $x\in E$ so that for each integer $k\geq2$, the set of "$k$-chains" with initial point at $x$ has positive Lebesgue measure. The second problem is a continuous analogue of the Erdős unit distance p…

多线性广义Radon变换与点构型

Multilinear generalized Radon transforms and point configurations

我们研究多线性广义Radon变换使用图论范式，包括广泛研究的线性情况。这些提供了一个通用的机制来研究Falconer型问题，涉及$(k+1)$ -点配置的几何测量理论，与$k \ge 2$，包括分布的单形，体积和角度确定的分形子集$E \subset {\Bbb R}^d$，$d \ge 2$的点。如果$T_k(E)$表示由$E$确定的非全等$(k+1)$ -点构型的集合，我们证明了如果$E$的Hausdorff维数大于$d-\frac{d-1}{2k}$，则$T_k(E)$的${k+1 \choose 2}$ -维Lebesgue测度是正的.这是对以前关于法尔科纳猜想的工作（\cite{Erd05}和那里的参考文献）以及关于有限点配置\cite{EHI11,GI10}的工作的补充。我们也给应用程序的离散几何和分形正则值定理的Erdös型问题，提供了一个多线性框架的结果在\cite{EIT11}。

实线上分形集的曲率三点模式问题

A curved three-point pattern problem for fractal sets on the real line

研究实线上分形集的曲率三点配置出现，证明当分形维数足够大时，集内存在与非线性函数相关联的曲率三点进展。